Furthermore, as an extension of Zakharov's idea, a Hamiltonian formulation for nonlinear water waves
with constant vorticity has been derived by [1]} (see also [2]}).
The associated symplectic structure in terms of \((\eta ,\xi )\)  is not canonical but a change of variables reduces the system to a canonical one.
Based on this formulation, recent work has been conducted involving long-wave modeling in the Korteweg–de Vries regime [3]},
rigorous mathematical analysis on the existence of quasi-periodic traveling wave solutions [4]},
and direct numerical simulation of unsteady waves on deep or shallow water [5]}, [6]}.
In particular, the reduction to surface variables makes it possible to solve the full equations
via efficient and accurate numerical solvers such as the boundary integral method, the conformal mapping technique
or the high-order spectral method.
